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Aaron Mazel-Gee
Aaron Mazel-Gee
1:42 PM
@AdrianClough @HarryGindi @RuneHaugseng thanks for your input [regarding presheaves of categories]. i've concluded that there doesn't seem to be any direct formula like i was asking for. it feels like right fibrations should be generalized to something slightly different from cartesian fibrations for something like this to hold.
 
Aaron Mazel-Gee
Aaron Mazel-Gee
1:57 PM
suppose i have a group $G$ equipped with a presentation. suppose i consider that as a presentation of an $\infty$-group $\tilde{G}$. then i get a map $\tilde{G} \to G$. is anyone aware of general conditions under which that map is an equivalence?
@lemiller that paper of pstragowski--vankoughnett is very nice, and if it works for your purposes you should certainly use it. however, if you aren't working in a context where you have a periodicity operator, then you may want to check out the version of goerss--hopkins that i worked out, which is basically just a direct generalization of the original one.
 
S. carmeli
S. carmeli
2:22 PM
I have a very silly names convention question: if I have a cone f_i:X-->Y_i, how should I call the induced map to the limit? is there a standard name for it?
 
Piotr Pstrągowski
Piotr Pstrągowski
If an $\infty$-category $C$ admits geometric realizations (equivalently, totalizations of cosimplicial objects), is it necessarily idempotent-complete? Lurie shows in HTT that this is the case when $C$ has filtered colimits.
 
S. carmeli
S. carmeli
@PiotrPstrągowski I would guess that the category of bounded below complexes over an additive, non idempotent complete, category, will be a counter example. Geometric realizations are then just passing to the total complex, but in general it shouldn't be idempotent complete (even if A is Im not sure it follows).
of course Im not sure but I would try to look at this example
 
Piotr Pstrągowski
Piotr Pstrągowski
2:44 PM
@S.carmeli The bounded version of this example is that Lurie uses to show that finite limits and colimits is not enough. What I worry about is that, for example, we can take bounded below complexes of free modules. But any finitely generated projective $P$ can be in fact represented by a non-bounded complex of free modules, say $... \rightarrow R^{n} \rightarrow R^{n} \rightarrow R^{n} \rightarrow 0$ where $R^{n} \rightarrow P$ is some surjection.
Moreover, in any ordinary category $C$, if $e: X \rightarrow X$ is an idempotent, then its splitting can be obtained by taking the coequalizer of $e$ and $id_{X}$. This suggests that perhaps in the $\infty$-category case there is some semisimplicial object whose colimit would split a given idempotent, but perhaps that's note quite how it goes?
 
S. carmeli
S. carmeli
@PiotrPstrągowski if it do it would make me extremely happy actually. For some reason I thought that the category of bounded below complexes of Frechet topological vector spaces is not, but I might have been wrong about it. Knowing it is the case whould make me personally much happier so tell me if you discover it works :-)
 
 
1 hour later…
lemiller
lemiller
3:58 PM
@AaronMazel-Gee Thanks, I've been enjoying your paper. One thing from the original one is a proof that the relative cotangent complex corepresents Der, which matches the situation I'm looking at. However, this is based on the exact triangle and I didn't see quickly in your paper a statement that really pinned down (it could take some translation) a positive answer for me. I also saw in a text that Toen asserts the non-relative version, namely that for sAlg_k, L_A corepresents RDer.
(I think Goerss-Hopkins also meant RDer), but I don't see it in G-H. He also quotes something by Quillen as a source, does anyone know for sure if this non-relative version holds? Or if I replace Der(B,-)/Omega_B with another representable functor F_B with functorial representing object T_B on Alg_k, if on the nose, the extension F_B to sAlg_k is representable by L T_B for the left Kan extension (assuming the context is such that the syntax makes sense)?
 
 
8 hours later…
Saal Hardali
Saal Hardali
11:32 PM
@PiotrPstrągowski @S.carmeli Let J be a category with a weakly terminal object X and consider the augmented simplicial diagram $y(X)^{\times \bullet + 1}$. Since X is weakly terminal this diagram is the cech nerve of an effective epimorphism in Psh(J) and therefore a colimit diagram.
It follows that any category which admits finite products and geometric realizations has colimits for all diagrams with a weakly terminal object (this is an argument I learned from you Piotr in this very chat room ^^).
So if haven't missed anything this seems to apply that at the very least if there are finite products (for example if the category is additive) and geometric realizations you also have retracts...
Sorry if I'm telling you both stuff you know already...
 

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